Positive Lower-Bounded Distributions
The positive lower-bounded probabilities have support on real values above some positive minimum value.
Pareto distribution
Probability density function
If \(y_{\text{min}} \in \mathbb{R}^+\) and \(\alpha \in \mathbb{R}^+\), then for \(y \in \mathbb{R}^+\) with \(y \geq y_{\text{min}}\), \[\begin{equation*} \text{Pareto}(y|y_{\text{min}},\alpha) = \frac{\displaystyle \alpha\,y_{\text{min}}^\alpha}{\displaystyle y^{\alpha+1}}. \end{equation*}\]
Distribution statement
y ~ pareto(y_min, alpha)
Increment target log probability density with pareto_lupdf(y | y_min, alpha).
Stan functions
real pareto_lpdf(reals y | reals y_min, reals alpha)
The log of the Pareto density of y given positive minimum value y_min and shape alpha
real pareto_lupdf(reals y | reals y_min, reals alpha)
The log of the Pareto density of y given positive minimum value y_min and shape alpha dropping constant additive terms
real pareto_cdf(reals y | reals y_min, reals alpha)
The Pareto cumulative distribution function of y given positive minimum value y_min and shape alpha
real pareto_lcdf(reals y | reals y_min, reals alpha)
The log of the Pareto cumulative distribution function of y given positive minimum value y_min and shape alpha
real pareto_lccdf(reals y | reals y_min, reals alpha)
The log of the Pareto complementary cumulative distribution function of y given positive minimum value y_min and shape alpha
R pareto_rng(reals y_min, reals alpha)
Generate a Pareto variate with positive minimum value y_min and shape alpha; may only be used in transformed data and generated quantities blocks. For a description of argument and return types, see section vectorized PRNG functions.
Pareto type 2 distribution
Probability density function
If \(\mu \in \mathbb{R}\), \(\lambda \in \mathbb{R}^+\), and \(\alpha \in \mathbb{R}^+\), then for \(y \geq \mu\), \[\begin{equation*} \mathrm{Pareto\_Type\_2}(y|\mu,\lambda,\alpha) = \ \frac{\alpha}{\lambda} \, \left( 1+\frac{y-\mu}{\lambda} \right)^{-(\alpha+1)} \! . \end{equation*}\]
Note that the Lomax distribution is a Pareto Type 2 distribution with \(\mu=0\).
Distribution statement
y ~ pareto_type_2(mu, lambda, alpha)
Increment target log probability density with pareto_type_2_lupdf(y | mu, lambda, alpha).
Stan functions
real pareto_type_2_lpdf(reals y | reals mu, reals lambda, reals alpha)
The log of the Pareto Type 2 density of y given location mu, scale lambda, and shape alpha
real pareto_type_2_lupdf(reals y | reals mu, reals lambda, reals alpha)
The log of the Pareto Type 2 density of y given location mu, scale lambda, and shape alpha dropping constant additive terms
real pareto_type_2_cdf(reals y | reals mu, reals lambda, reals alpha)
The Pareto Type 2 cumulative distribution function of y given location mu, scale lambda, and shape alpha
real pareto_type_2_lcdf(reals y | reals mu, reals lambda, reals alpha)
The log of the Pareto Type 2 cumulative distribution function of y given location mu, scale lambda, and shape alpha
real pareto_type_2_lccdf(reals y | reals mu, reals lambda, reals alpha)
The log of the Pareto Type 2 complementary cumulative distribution function of y given location mu, scale lambda, and shape alpha
R pareto_type_2_rng(reals mu, reals lambda, reals alpha)
Generate a Pareto Type 2 variate with location mu, scale lambda, and shape alpha; may only be used in transformed data and generated quantities blocks. For a description of argument and return types, see section vectorized PRNG functions.
Wiener First Passage Time Distribution
For an extended explanation of how to use the wiener_lpdf and wiener_l[c]cdf_unnorm functions, see Henrich et al. (2024).
Probability density function
If \(\alpha \in \mathbb{R}^+\), \(\tau \in \mathbb{R}^+\), \(\beta \in (0, 1)\), \(\delta \in \mathbb{R}\), \(s_{\delta} \in \mathbb{R}^{\geq 0}\), \(s_{\beta} \in [0, 1)\), and \(s_{\tau} \in \mathbb{R}^{\geq 0}\) then for \(y > \tau\),
\[\begin{equation*} \begin{split} &\text{Wiener}(y\mid \alpha,\tau,\beta,\delta,s_{\delta},s_{\beta},s_{\tau}) = \\ &\frac{1}{s_{\tau}}\int_{\tau}^{\tau+s_{\tau}}\frac{1}{s_{\beta}}\int_{\beta-\frac{1}{2}s_{\beta}}^{\beta+\frac{1}{2}s_{\beta}}\int_{-\infty}^{\infty} p_3(y-{\tau_0}\mid \alpha,\nu,\omega) \\ &\times \frac{1}{\sqrt{2\pi s_{\delta}^2}}\exp\Bigl(-\frac{(\nu-\delta)^2}{2s_{\delta}^2}\Bigr) \,d\nu \,d\omega \,d{\tau_0}= \\ &\frac{1}{s_{\tau}}\int_{\tau}^{\tau+s_{\tau}}\frac{1}{s_{\beta}}\int_{\beta-\frac{1}{2}s_{\beta}}^{\beta+\frac{1}{2}s_{\beta}} M\times p_3(y-{\tau_0}\mid \alpha,\nu,\omega) \,d\omega \,d{\tau_0}, \end{split} \end{equation*}\]
where \(p()\) denotes the density function, and \(M\) and \(p_3()\) are defined, by using \(t:=y-{\tau_0}\), as
\[\begin{equation*} M \coloneqq \frac{1}{\sqrt{1+s_{\delta}^2t}}\exp\Bigl(\alpha{\delta}\omega+\frac{\delta^2t}{2}+\frac{s_{\delta}^2\alpha^2\omega^2-2\alpha{\delta}\omega-\delta^2t}{2(1+s_{\delta}^2t)}\Bigr)\text{ and} \end{equation*}\]
\[\begin{equation*} p_3(t\mid \alpha,\delta,\beta) \coloneqq \frac{1}{\alpha^2}\exp\Bigl(-\alpha\delta\beta-\frac{\delta^2t}{2}\Bigr)f(\frac{t}{\alpha^2}\mid 0,1,\beta), \end{equation*}\]
where \(f(t^*=\frac{t}{\alpha^2}\mid0,1,\beta)\) can be specified in two ways:
\[\begin{equation*} f_l(t^*\mid 0,1,\beta) = \sum_{k=1}^\infty k\pi \exp\Bigl(-\frac{k^2\pi^2t^*}{2}\Bigr)\sin(k\pi \beta)\text{ and} \end{equation*}\]
\[\begin{equation*} f_s(t^*\mid0,1,\beta) = \sum_{k=-\infty}^\infty \frac{1}{\sqrt{2\pi(t^*)^3}}(\beta+2k) \exp\Bigl(-\frac{(\beta+2k)^2}{2t^*}\Bigr). \end{equation*}\]
Which of these is used in the computations depends on which expression requires the smaller number of components \(k\) to guarantee a pre-specified precision
In the case where \(s_{\delta}\), \(s_{\beta}\), and \(s_{\tau}\) are all \(0\), this simplifies to one representation that converges fast for small reaction-time values (“small time expansion”): \[\begin{equation*} \text{Wiener}(y|\alpha, \tau, \beta, \delta) = \frac{\alpha}{(y-\tau)^{3/2}} \exp \! \left(- \delta \alpha \beta - \frac{\delta^2(y-\tau)}{2}\right) \sum_{k = - \infty}^{\infty} (2k + \beta) \phi \! \left(\frac{(2k + \beta)\alpha }{\sqrt{y - \tau}}\right), \end{equation*}\] where \(\phi(x)\) denotes the standard normal density function, and one representation that converges fast for large reaction-time values (“large time expansion”): \[\begin{equation*} \text{Wiener}(y|\alpha, \tau, \beta, \delta) = \frac{\pi}{\alpha^2} \exp \! \left(- \delta \alpha \beta - \frac{\delta^2(y-\tau)}{2}\right) \sum_{k = 1}^{\infty} k \exp \! \left(-\frac{k^2\pi^2(y-\tau)}; {2\alpha^2}\right) \sin \!(k\pi\beta) \end{equation*}\] see (Feller 1968), (Navarro and Fuss 2009).
Cumulative distribution function
For the cumulative distribution function (cdf) there also exist two expressions depending on the reaction time.
Let \(\alpha\), \(\tau\), \(\beta\), \(\delta\), \(s_{\delta}\), \(s_{\beta}\), \(s_{\tau}\) and \(y\) be as above.
The formula for the large-time cdf of decision times (excluding the additive reaction time components summarized in \(\tau\) for the time being) at the upper boundary is stated as follows:
\[\begin{equation} F(y|\alpha, \beta, \delta) = P(\alpha, \beta, \delta) - \exp\left(\delta\alpha(1-\beta)-\frac{\delta^2 y}{2}\right)F_l(y|\alpha,\beta,\delta), \end{equation}\] where \(P(\alpha,\beta,\delta)\) is the probability to hit the upper boundary, defined as
\[\begin{equation} P(\alpha, \beta, \delta) = \begin{cases} \frac{1-\exp(2\delta \alpha \beta)}{\exp(-2\delta \alpha(1-\beta)) - \exp(2\delta \alpha \beta)}, & \text{for } \delta\neq 0 \\ \beta, & \text{for } \delta=0, \end{cases} \end{equation}\]
and
\[\begin{equation} F_l(y|\alpha, \beta, \delta) = \frac{2\pi}{\alpha^2}\sum_{k=1}^{\infty}{\frac{k\sin{k\pi(1-\beta)}}{\delta^2+(k\pi)^2/\alpha^2}}\exp(-\frac{k^2\pi^2y}{2\alpha^2}). \end{equation}\]
The formula for the small-time cdf at the upper boundary is stated as follows:
\[\begin{equation} F(y|\alpha,\beta,\delta) = \exp\left(\delta \alpha(1-\beta) -\frac{\delta^2y}{2}\right)F_s(y|\alpha, \beta,\delta), \end{equation}\] where
\[\begin{equation} F_s(y|\alpha,\beta,\delta) = \sum_{k=0}^{\infty}(-1)^k\phi\left(\frac{\alpha(k+\beta^{*}_k)} {\sqrt{y}} \right) \times \left( R \left(\frac{\alpha(k+\beta^{*}_k)+\delta y}{\sqrt{y}} \right) + R \left(\frac{\alpha(k+\beta^*_k)-\delta y}{\sqrt{y}} \right)\right), \end{equation}\]
where \(\beta^*_k=(1-\beta)\) for \(k\) even, \(\beta^*_k=\beta\) for \(k\) odd, and \(R\) is Mill’s ratio.
The cdf for the lower boundary is \(F(y|\alpha,1-\beta,-\delta)\)
Distribution statement
y ~ wiener(alpha, tau, beta, delta)
Increment target log probability density with wiener_lupdf(y | alpha, tau, beta, delta).
y ~ wiener(alpha, tau, beta, delta, var_delta) Increment target log probability density with wiener_lupdf(y | alpha, tau, beta, delta, var_delta).
y ~ wiener(alpha, tau, beta, delta, var_delta, var_beta, var_tau) Increment target log probability density with wiener_lupdf(y | alpha, tau, beta, delta, var_delta, var_beta, var_tau).
Stan functions
real wiener_lpdf(reals y | reals alpha, reals tau, reals beta, reals delta)
The log of the Wiener first passage time density of y given boundary separation alpha, non-decision time tau, starting point beta, and drift rate delta.
real wiener_lpdf(real y | real alpha, real tau, real beta, real delta, real var_delta)
The log of the Wiener first passage time density of y given boundary separation alpha, non-decision time tau, starting point beta, drift rate delta, and inter-trial drift rate variability var_delta.
Setting var_delta to 0 recovers the 4-parameter signature above.
real wiener_lpdf(real y | real alpha, real tau, real beta, real delta, real var_delta, real var_beta, real var_tau)
The log of the Wiener first passage time density of y given boundary separation alpha, non-decision time tau, starting point beta, drift rate delta, inter-trial drift rate variability var_delta, inter-trial variability of the starting point (bias) var_beta, and inter-trial variability of the non-decision time var_tau.
Setting var_delta, var_beta, and var_tau to 0 recovers the 4-parameter signature above.
real wiener_lupdf(reals y | reals alpha, reals tau, reals beta, reals delta)
The log of the Wiener first passage time density of y given boundary separation alpha, non-decision time tau, starting point beta, and drift rate delta, dropping constant additive terms
real wiener_lupdf(real y | real alpha, real tau, real beta, real delta, real var_delta)
The log of the Wiener first passage time density of y given boundary separation alpha, non-decision time tau, starting point beta, drift rate delta, and inter-trial drift rate variability var_delta, dropping constant additive terms.
Setting var_delta to 0 recovers the 4-parameter signature above.
real wiener_lupdf(real y | real alpha, real tau, real beta, real delta, real var_delta, real var_beta, real var_tau)
The log of the Wiener first passage time density of y given boundary separation alpha, non-decision time tau, starting point beta, drift rate delta, inter-trial drift rate variability var_delta, inter-trial variability of the starting point (bias) var_beta, and inter-trial variability of the non-decision time var_tau, dropping constant additive terms.
Setting var_delta, var_beta, and var_tau to 0 recovers the 4-parameter signature above.
Note: The lcdf and lccdf functions for the wiener distribution are conditional and unnormalized, meaning that the cdf does not asymptote at 1, but rather at the probability to hit the upper boundary.
Similarly, the ccdf is defined as the probability to hit the upper boundary less the value of the cdf, as opposed to the more typical \(1 - \textrm{cdf}\).
real wiener_lcdf_unnorm(real y, real alpha, real tau, real beta, real delta)
The log of the cumulative distribution function (cdf) of the Wiener distribution of y given boundary separation alpha, non-decision time tau, starting point beta, and drift rate delta.
real wiener_lccdf_unnorm(real y, real alpha, real tau, real beta, real delta)
The log of the complementary cumulative distribution function (ccdf) of the Wiener distribution of y given boundary separation alpha, non-decision time tau, starting point beta, and drift rate delta.
real wiener_lcdf_unnorm(real y, real alpha, real tau, real beta, real delta, real var_delta, real var_beta, real var_tau)
The log of the cumulative distribution function (cdf) of the Wiener distribution of y given boundary separation alpha, non-decision time tau, starting point beta, drift rate delta, inter-trial drift rate variability var_delta, inter-trial variability of the starting point (bias) var_beta, and inter-trial variability of the non-decision time var_tau.
real wiener_lccdf_unnorm(real y, real alpha, real tau, real beta, real delta, real var_delta, real var_beta, real var_tau)
The log of the complementary cumulative distribution function (ccdf) of the Wiener distribution of y given boundary separation alpha, non-decision time tau, starting point beta, drift rate delta, inter-trial drift rate variability var_delta, inter-trial variability of the starting point (bias) var_beta, and inter-trial variability of the non-decision time var_tau.
Boundaries
Stan returns the first passage time of the accumulation process over the upper boundary only. To get the result for the lower boundary, use \[\begin{equation*} \text{Wiener}(y | \alpha, \tau, 1 - \beta, - \delta) \end{equation*}\] For more details, see the appendix of Vandekerckhove and Wabersich (2014).
Vectorization
The 5- and 7-argument forms of the wiener distribution functions (listed above as recieving only real) are implemented in such a way where they can be fully vectorized, but currently only versions that accept all real and all vector arguments are exposed by Stan. If there are additional signatures that would prove useful, please request them by opening an issue.
Tolerance tuning
The 5- and 7-argument forms of the wiener distribution functions can also accept an additional data real argument controlling the required precision of the gradient calculation of the function. If omitted, this defaults to 1e-4 for the density and 1e-8 for the cdf functions.